## Generation of oriented matroids -- A graph theoretical approach (2002)

Venue: | DISCRETE COMPUT GEOM |

Citations: | 9 - 2 self |

### BibTeX

@ARTICLE{Finschi02generationof,

author = {Lukas Finschi and Komei Fukuda},

title = {Generation of oriented matroids -- A graph theoretical approach },

journal = {DISCRETE COMPUT GEOM},

year = {2002},

volume = {27},

pages = {117--136}

}

### OpenURL

### Abstract

We discuss methods for the generation of oriented matroids and of isomorphism classes of oriented matroids. Our methods are based on single element extensions and graph theoretical representations of oriented matroids, and all these methods work in general rank and for nonuniform and uniform oriented matroids as well. We consider two types of graphs, cocircuit graphs and tope graphs, and discuss the single element extensions in terms of localizations which can be viewed as partitions of the vertex sets of the graphs. Whereas localizations of the cocircuit graph are well characterized, there is no graph theoretical characterization known for localizations of the tope graph. In this paper we prove a connectedness property for tope graph localizations and use this for the design of algorithms for the generation of single element extensions by use of tope graphs. Furthermore we discuss similar algorithms which use the cocircuit graph. The characterization of localizations of cocircuit graphs finally leads to a backtracking algorithm which is a simple and efficient method for the generation of single element extensions. We compare this method with a recent algorithm of Bokowski and Guedes de Oliveira for uniform oriented matroids.

### Citations

134 |
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(Show Context)
Citation Context ... has connections and applications to many areas of mathematics. For a comprehensive introduction to the theory of OMs we refer to the monograph of Bjorner, Las Vergnas, Sturmfels, White, and Ziegler [=-=4]-=-. We will use in the followingsnite sphere arrangements as an illustration of OMs. Asnite sphere arrangement S = fS e j e 2 Eg is a collection of (d 1)-dimensional unit spheres on the ddimensional uni... |

62 |
Oriented Matroids
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- 1978
(Show Context)
Citation Context ... real hyperplane arrangements, convex polytopes, or point congurations in the Euclidean space. The notion of OMs was introduced independently by Bland and Las Vergnas [6] and by Folkman and Lawrence [=-=13-=-]. There are several dierent (but equivalent) axiom systems and representations of OMs, and the theory of OMs has connections and applications to many areas of mathematics. For a comprehensive introdu... |

46 |
Las Vergnas, Orientability of Matroids
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- 1978
(Show Context)
Citation Context ... of geometric structures such as real hyperplane arrangements, convex polytopes, or point congurations in the Euclidean space. The notion of OMs was introduced independently by Bland and Las Vergnas [=-=6-=-] and by Folkman and Lawrence [13]. There are several dierent (but equivalent) axiom systems and representations of OMs, and the theory of OMs has connections and applications to many areas of mathema... |

23 |
Hyperplane arrangements with a lattice of regions. Discrete & computational geometry
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- 1990
(Show Context)
Citation Context ...n generating single element extensions by introducing loops or parallel elements: these extensions can be considered as being trivial. The tope graph of a (simple) OM determines its isomorphism class =-=[3]-=-, and there are ecient algorithms for computing a representative OM from the given tope graph [10]. Furthermore, it is possible to decide for a given graph in polynomial time whether it is a tope grap... |

20 |
Oriented matroid programming
- Fukuda
- 1982
(Show Context)
Citation Context ...4). We summarize the results on OM programming needed for the proof of Theorem 3.5 in the following theorem, which is essentially the strong duality theorem for OM programming [5, 13, 4]: 3.6 Theorem =-=[14]-=- For any OM program (M; g; f), which is a triple of an OM M = (E; F) and two distinct elements f; g 2 E, exactly one of the following three statements is valid: (i) (M; g; f) is not feasible, i.e. the... |

18 |
A combinatorial abstraction of linear programming
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(Show Context)
Citation Context ...ected in the sense of (L4). We summarize the results on OM programming needed for the proof of Theorem 3.5 in the following theorem, which is essentially the strong duality theorem for OM programming =-=[5, 13, 4]-=-: 3.6 Theorem [14] For any OM program (M; g; f), which is a triple of an OM M = (E; F) and two distinct elements f; g 2 E, exactly one of the following three statements is valid: (i) (M; g; f) is not ... |

11 |
Antipodal graphs and oriented matroids
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(Show Context)
Citation Context ...re are ecient algorithms for computing a representative OM from the given tope graph [10]. Furthermore, it is possible to decide for a given graph in polynomial time whether it is a tope graph or not =-=[15, 16]-=-. So, isomorphism classes of simple OMs can be generated if it is possible to generate tope graphs or at least a (not too large) superset of graphs. Unfortunately, the known characterizations of OM to... |

10 |
Polytopal and nonpolytopal spheres. An algorithmic approach
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- 1987
(Show Context)
Citation Context ...or eciently generating OMs leads to new results for OM representations. Techniques for listing OMs for small d and jEj were studied, among others, by Bokowski, Sturmfels, and Guedes de Oliveira (e.g. =-=[7, 8, 9]-=-) using the chirotope axioms of OMs. They also proved by successful applications e.g. to geometric embeddability problems the usefulness of OM generation. However, it seems that the methods are design... |

8 |
de Oliveira, On the generation of oriented matroids
- Bokowski, Guedes
- 2000
(Show Context)
Citation Context ...or eciently generating OMs leads to new results for OM representations. Techniques for listing OMs for small d and jEj were studied, among others, by Bokowski, Sturmfels, and Guedes de Oliveira (e.g. =-=[7, 8, 9]-=-) using the chirotope axioms of OMs. They also proved by successful applications e.g. to geometric embeddability problems the usefulness of OM generation. However, it seems that the methods are design... |

6 |
Fukuda: Reverse search for enumeration
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- 1996
(Show Context)
Citation Context ...d as it makes it possible to generate all weak acycloidal signatures of G without repetition. For this we modify a reverse search method for the generation of all connected subgraphs of a given graph =-=-=-[1]. Enumerate the vertices of the given tope graph G in an arbitrary way such that V (G) = f1; : : : ; ng. Remember that every weak acycloidal signature denes a set V := fv 2 V (G) j (v) = g, and th... |

6 |
Combinatorial face enumeration in arrangements and oriented matroids, Discrete Applied Mathematics 31
- Fukuda, Saito, et al.
- 1991
(Show Context)
Citation Context ...re are ecient algorithms for computing a representative OM from the given tope graph [10]. Furthermore, it is possible to decide for a given graph in polynomial time whether it is a tope graph or not =-=[15, 16]-=-. So, isomorphism classes of simple OMs can be generated if it is possible to generate tope graphs or at least a (not too large) superset of graphs. Unfortunately, the known characterizations of OM to... |

6 |
Vergnas, Extensions Ponctuelles d’une Géométrie Combinatoire Orienté, in “Problèmes Combinatoires et Théorie des Graphes
- Las
- 1976
(Show Context)
Citation Context ...be a coline. The edges in E(G) whose coline is U form a cycle c(U) in G which we call the coline cycle of U . The following characterization of localizations of cocircuit graphs is due to Las Vergnas =-=[18, -=-4]: 5.2 Theorem Let G be the cocircuit graph of an OM M, given with the set of all coline cycles of G, and : V (G) ! f+; 0; g a signature of G. Then: is a localization of G w.r.t. M if and only if f... |

5 | Oriented matroids and combinatorial manifolds - Cordovil, Fukuda - 1993 |

4 |
de Oliveira, On the cocircuit graph of an oriented matroid, Discrete Comput
- Cordovil, Fukuda, et al.
(Show Context)
Citation Context ...ection 2). Again we are interested in simple OMs and may restrict our discussion accordingly. In contrast to tope graphs, a cocircuit graph of an OM M does not characterize the isomorphism class of M =-=[11-=-], and there is also no characterization of cocircuit graphs known. Nevertheless cocircuit graphs will be helpful, as shown in the following; a major benet comes from a characterization of localizatio... |

3 | A Characterization of oriented matroids in terms of topes - Handa - 1990 |

2 |
Fukuda: Cocircuit graphs and ecient orientation reconstruction in oriented matroids
- Babson, Finschi, et al.
- 2001
(Show Context)
Citation Context ...et G be the cocircuit graph of some OM and : V (G) ! f+; 0; g a localization of G. Then: G + (and also G ) is a connected subgraph of G. For the proof of the lemma we need the following result from [=-=11, 2]-=-: 5.4 Lemma Let M = (E; F) be an OM with cocircuit graph G and associating bijection L : V (G) ! C. For an arbitrary element e 2 E let V + e denote the set of vertices with L(v) e = +. Then: The subgr... |

2 | Axioms for maximal vectors of an oriented matroid: a combinatorial characterization of the regions determined by an arrangement of pseudohyperplanes, Europ - Silva - 1995 |